What is interesting for students to find out is what the effect of a change in
a, b, or c will have on the graph of a parabola.
Start with a given equation where a, b and c are known. Have students draw several
parabolas by keeping a and b as given, and change c. They should see that for some
values of c there are no roots, one root, or two roots. They should also notice
that the shape of the parabola stays the same (congruent) and that the y-intercept
and vertex changes. For those equations that have roots, they can be found by
using the CALC menu.
Similarly, keep b and c as given and change the values of a for a both positive
and negative, as well as larger than one and between zero and one. Students should
see that the vertex does not remain the same, the y-intercept changes, and that
the number of roots as well as the value of roots changes.
Similarly, keep a and c as given and change the values of b. When the values of b
change in sequence, the movement of the parabola's vertex is the path of another
parabola. The vertex changes, the y-intercept remains the same but the number and
value of the roots change.
You mentioned that you had a TI-82. I am using a TI-83. The 82 may have the same
feature. Try it. In your y= menu enter an equation:
y = 2x^2 - 5x + { -3, -2, -1, 0, 1, 2, 3 }
The seven graphs will be drawn in sequence. This is where I believe that the
graphing calculator is so meaningful for instruction. Similarly, replace the
value of 2 with a set of numbers such as
{ -4, -2, -1, -0.5, 0, 0.25, 0.5, 1, 2, 4 }
and keep one of the c values. The motion of the graphs is great!
Hope that this gives you some ideas.